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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380.179"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.18
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.18

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.784760 + 1.17650i) q^{2} +(-2.02176 - 1.16726i) q^{3} +(-0.768302 - 1.84654i) q^{4} +(1.88679 - 1.20001i) q^{5} +(2.95988 - 1.46258i) q^{6} -0.943490 q^{7} +(2.77539 + 0.545184i) q^{8} +(1.22501 + 2.12179i) q^{9} +(-0.0688652 + 3.16153i) q^{10} +0.173788i q^{11} +(-0.602077 + 4.63008i) q^{12} +(-0.200082 - 0.346553i) q^{13} +(0.740414 - 1.11002i) q^{14} +(-5.21537 + 0.223754i) q^{15} +(-2.81942 + 2.83740i) q^{16} +(-2.08538 - 1.20400i) q^{17} +(-3.45762 - 0.223866i) q^{18} +(-4.32892 - 0.510313i) q^{19} +(-3.66549 - 2.56206i) q^{20} +(1.90751 + 1.10130i) q^{21} +(-0.204461 - 0.136382i) q^{22} +(-3.88561 - 6.73008i) q^{23} +(-4.97480 - 4.34184i) q^{24} +(2.11995 - 4.52834i) q^{25} +(0.564736 + 0.0365641i) q^{26} +1.28393i q^{27} +(0.724885 + 1.74219i) q^{28} +(-4.11985 + 2.37860i) q^{29} +(3.82957 - 6.31147i) q^{30} +6.51703 q^{31} +(-1.12563 - 5.54373i) q^{32} +(0.202856 - 0.351357i) q^{33} +(3.05303 - 1.50860i) q^{34} +(-1.78017 + 1.13220i) q^{35} +(2.97678 - 3.89221i) q^{36} -9.07290 q^{37} +(3.99755 - 4.69250i) q^{38} +0.934196i q^{39} +(5.89080 - 2.30185i) q^{40} +(-3.80514 - 2.19690i) q^{41} +(-2.79262 + 1.37993i) q^{42} +(0.622036 - 1.07740i) q^{43} +(0.320906 - 0.133521i) q^{44} +(4.85751 + 2.53333i) q^{45} +(10.9672 + 0.710078i) q^{46} +(-3.10145 - 5.37186i) q^{47} +(9.01220 - 2.44554i) q^{48} -6.10983 q^{49} +(3.66393 + 6.04778i) q^{50} +(2.81076 + 4.86839i) q^{51} +(-0.486200 + 0.635718i) q^{52} +(1.49617 + 2.59144i) q^{53} +(-1.51054 - 1.00757i) q^{54} +(0.208547 + 0.327901i) q^{55} +(-2.61855 - 0.514376i) q^{56} +(8.15638 + 6.08473i) q^{57} +(0.434677 - 6.71363i) q^{58} +(-4.78221 + 8.28302i) q^{59} +(4.42015 + 9.45848i) q^{60} +(-3.58110 - 6.20264i) q^{61} +(-5.11430 + 7.66728i) q^{62} +(-1.15579 - 2.00188i) q^{63} +(7.40555 + 3.02620i) q^{64} +(-0.793380 - 0.413771i) q^{65} +(0.254178 + 0.514391i) q^{66} +(9.56423 - 5.52191i) q^{67} +(-0.621023 + 4.77577i) q^{68} +18.1422i q^{69} +(0.0649736 - 2.98287i) q^{70} +(-4.54516 + 7.87245i) q^{71} +(2.24312 + 6.55664i) q^{72} +(7.54872 + 4.35826i) q^{73} +(7.12006 - 10.6743i) q^{74} +(-9.57180 + 6.68068i) q^{75} +(2.38361 + 8.38561i) q^{76} -0.163967i q^{77} +(-1.09908 - 0.733120i) q^{78} +(4.94885 - 8.57166i) q^{79} +(-1.91474 + 8.73692i) q^{80} +(5.17372 - 8.96115i) q^{81} +(5.57077 - 2.75271i) q^{82} -5.08393 q^{83} +(0.568054 - 4.36843i) q^{84} +(-5.37948 + 0.230795i) q^{85} +(0.779409 + 1.57732i) q^{86} +11.1058 q^{87} +(-0.0947463 + 0.482328i) q^{88} +(3.34144 - 1.92918i) q^{89} +(-6.79245 + 3.72680i) q^{90} +(0.188776 + 0.326969i) q^{91} +(-9.44204 + 12.3457i) q^{92} +(-13.1759 - 7.60709i) q^{93} +(8.75389 + 0.566775i) q^{94} +(-8.78015 + 4.23190i) q^{95} +(-4.19524 + 12.5220i) q^{96} +(4.28716 - 7.42557i) q^{97} +(4.79475 - 7.18821i) q^{98} +(-0.368740 + 0.212892i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.784760 + 1.17650i −0.554909 + 0.831911i
\(3\) −2.02176 − 1.16726i −1.16726 − 0.673921i −0.214230 − 0.976783i \(-0.568724\pi\)
−0.953034 + 0.302863i \(0.902058\pi\)
\(4\) −0.768302 − 1.84654i −0.384151 − 0.923270i
\(5\) 1.88679 − 1.20001i 0.843798 − 0.536661i
\(6\) 2.95988 − 1.46258i 1.20837 − 0.597095i
\(7\) −0.943490 −0.356606 −0.178303 − 0.983976i \(-0.557061\pi\)
−0.178303 + 0.983976i \(0.557061\pi\)
\(8\) 2.77539 + 0.545184i 0.981248 + 0.192752i
\(9\) 1.22501 + 2.12179i 0.408338 + 0.707262i
\(10\) −0.0688652 + 3.16153i −0.0217771 + 0.999763i
\(11\) 0.173788i 0.0523989i 0.999657 + 0.0261995i \(0.00834050\pi\)
−0.999657 + 0.0261995i \(0.991659\pi\)
\(12\) −0.602077 + 4.63008i −0.173805 + 1.33659i
\(13\) −0.200082 − 0.346553i −0.0554929 − 0.0961165i 0.836944 − 0.547288i \(-0.184340\pi\)
−0.892437 + 0.451171i \(0.851006\pi\)
\(14\) 0.740414 − 1.11002i 0.197884 − 0.296664i
\(15\) −5.21537 + 0.223754i −1.34660 + 0.0577729i
\(16\) −2.81942 + 2.83740i −0.704856 + 0.709351i
\(17\) −2.08538 − 1.20400i −0.505779 − 0.292012i 0.225318 − 0.974285i \(-0.427658\pi\)
−0.731097 + 0.682273i \(0.760991\pi\)
\(18\) −3.45762 − 0.223866i −0.814970 − 0.0527656i
\(19\) −4.32892 − 0.510313i −0.993123 − 0.117074i
\(20\) −3.66549 − 2.56206i −0.819629 − 0.572894i
\(21\) 1.90751 + 1.10130i 0.416253 + 0.240324i
\(22\) −0.204461 − 0.136382i −0.0435912 − 0.0290767i
\(23\) −3.88561 − 6.73008i −0.810206 − 1.40332i −0.912719 − 0.408587i \(-0.866022\pi\)
0.102513 − 0.994732i \(-0.467312\pi\)
\(24\) −4.97480 − 4.34184i −1.01548 − 0.886275i
\(25\) 2.11995 − 4.52834i 0.423990 − 0.905667i
\(26\) 0.564736 + 0.0365641i 0.110754 + 0.00717081i
\(27\) 1.28393i 0.247092i
\(28\) 0.724885 + 1.74219i 0.136990 + 0.329243i
\(29\) −4.11985 + 2.37860i −0.765037 + 0.441694i −0.831101 − 0.556121i \(-0.812289\pi\)
0.0660643 + 0.997815i \(0.478956\pi\)
\(30\) 3.82957 − 6.31147i 0.699180 − 1.15231i
\(31\) 6.51703 1.17049 0.585246 − 0.810856i \(-0.300998\pi\)
0.585246 + 0.810856i \(0.300998\pi\)
\(32\) −1.12563 − 5.54373i −0.198985 − 0.980002i
\(33\) 0.202856 − 0.351357i 0.0353127 − 0.0611634i
\(34\) 3.05303 − 1.50860i 0.523590 − 0.258723i
\(35\) −1.78017 + 1.13220i −0.300903 + 0.191376i
\(36\) 2.97678 − 3.89221i 0.496131 − 0.648702i
\(37\) −9.07290 −1.49158 −0.745788 − 0.666183i \(-0.767927\pi\)
−0.745788 + 0.666183i \(0.767927\pi\)
\(38\) 3.99755 − 4.69250i 0.648488 − 0.761224i
\(39\) 0.934196i 0.149591i
\(40\) 5.89080 − 2.30185i 0.931417 − 0.363954i
\(41\) −3.80514 − 2.19690i −0.594263 − 0.343098i 0.172518 − 0.985006i \(-0.444810\pi\)
−0.766781 + 0.641908i \(0.778143\pi\)
\(42\) −2.79262 + 1.37993i −0.430911 + 0.212928i
\(43\) 0.622036 − 1.07740i 0.0948596 − 0.164302i −0.814690 − 0.579896i \(-0.803093\pi\)
0.909550 + 0.415595i \(0.136426\pi\)
\(44\) 0.320906 − 0.133521i 0.0483784 − 0.0201291i
\(45\) 4.85751 + 2.53333i 0.724115 + 0.377647i
\(46\) 10.9672 + 0.710078i 1.61703 + 0.104695i
\(47\) −3.10145 − 5.37186i −0.452392 − 0.783567i 0.546142 − 0.837693i \(-0.316096\pi\)
−0.998534 + 0.0541261i \(0.982763\pi\)
\(48\) 9.01220 − 2.44554i 1.30080 − 0.352983i
\(49\) −6.10983 −0.872832
\(50\) 3.66393 + 6.04778i 0.518158 + 0.855285i
\(51\) 2.81076 + 4.86839i 0.393586 + 0.681710i
\(52\) −0.486200 + 0.635718i −0.0674238 + 0.0881582i
\(53\) 1.49617 + 2.59144i 0.205514 + 0.355961i 0.950296 − 0.311347i \(-0.100780\pi\)
−0.744782 + 0.667308i \(0.767447\pi\)
\(54\) −1.51054 − 1.00757i −0.205558 − 0.137114i
\(55\) 0.208547 + 0.327901i 0.0281205 + 0.0442141i
\(56\) −2.61855 − 0.514376i −0.349918 − 0.0687364i
\(57\) 8.15638 + 6.08473i 1.08034 + 0.805942i
\(58\) 0.434677 − 6.71363i 0.0570759 − 0.881543i
\(59\) −4.78221 + 8.28302i −0.622590 + 1.07836i 0.366411 + 0.930453i \(0.380586\pi\)
−0.989002 + 0.147905i \(0.952747\pi\)
\(60\) 4.42015 + 9.45848i 0.570639 + 1.22108i
\(61\) −3.58110 − 6.20264i −0.458513 − 0.794167i 0.540370 − 0.841427i \(-0.318284\pi\)
−0.998883 + 0.0472605i \(0.984951\pi\)
\(62\) −5.11430 + 7.66728i −0.649517 + 0.973745i
\(63\) −1.15579 − 2.00188i −0.145616 − 0.252214i
\(64\) 7.40555 + 3.02620i 0.925693 + 0.378274i
\(65\) −0.793380 − 0.413771i −0.0984067 − 0.0513220i
\(66\) 0.254178 + 0.514391i 0.0312872 + 0.0633172i
\(67\) 9.56423 − 5.52191i 1.16846 − 0.674609i 0.215141 − 0.976583i \(-0.430979\pi\)
0.953316 + 0.301974i \(0.0976456\pi\)
\(68\) −0.621023 + 4.77577i −0.0753101 + 0.579148i
\(69\) 18.1422i 2.18406i
\(70\) 0.0649736 − 2.98287i 0.00776583 − 0.356521i
\(71\) −4.54516 + 7.87245i −0.539411 + 0.934288i 0.459524 + 0.888165i \(0.348020\pi\)
−0.998936 + 0.0461227i \(0.985313\pi\)
\(72\) 2.24312 + 6.55664i 0.264355 + 0.772707i
\(73\) 7.54872 + 4.35826i 0.883511 + 0.510095i 0.871814 − 0.489836i \(-0.162943\pi\)
0.0116966 + 0.999932i \(0.496277\pi\)
\(74\) 7.12006 − 10.6743i 0.827690 − 1.24086i
\(75\) −9.57180 + 6.68068i −1.10526 + 0.771418i
\(76\) 2.38361 + 8.38561i 0.273419 + 0.961895i
\(77\) − 0.163967i − 0.0186858i
\(78\) −1.09908 − 0.733120i −0.124446 − 0.0830095i
\(79\) 4.94885 − 8.57166i 0.556789 − 0.964387i −0.440973 − 0.897520i \(-0.645366\pi\)
0.997762 − 0.0668665i \(-0.0213002\pi\)
\(80\) −1.91474 + 8.73692i −0.214075 + 0.976817i
\(81\) 5.17372 − 8.96115i 0.574858 − 0.995684i
\(82\) 5.57077 − 2.75271i 0.615189 − 0.303986i
\(83\) −5.08393 −0.558034 −0.279017 − 0.960286i \(-0.590009\pi\)
−0.279017 + 0.960286i \(0.590009\pi\)
\(84\) 0.568054 − 4.36843i 0.0619798 − 0.476635i
\(85\) −5.37948 + 0.230795i −0.583487 + 0.0250332i
\(86\) 0.779409 + 1.57732i 0.0840458 + 0.170087i
\(87\) 11.1058 1.19067
\(88\) −0.0947463 + 0.482328i −0.0101000 + 0.0514163i
\(89\) 3.34144 − 1.92918i 0.354192 − 0.204493i −0.312338 − 0.949971i \(-0.601112\pi\)
0.666530 + 0.745478i \(0.267779\pi\)
\(90\) −6.79245 + 3.72680i −0.715987 + 0.392839i
\(91\) 0.188776 + 0.326969i 0.0197891 + 0.0342757i
\(92\) −9.44204 + 12.3457i −0.984401 + 1.28713i
\(93\) −13.1759 − 7.60709i −1.36627 − 0.788819i
\(94\) 8.75389 + 0.566775i 0.902895 + 0.0584584i
\(95\) −8.78015 + 4.23190i −0.900824 + 0.434184i
\(96\) −4.19524 + 12.5220i −0.428175 + 1.27802i
\(97\) 4.28716 − 7.42557i 0.435295 − 0.753953i −0.562025 − 0.827120i \(-0.689977\pi\)
0.997320 + 0.0731676i \(0.0233108\pi\)
\(98\) 4.79475 − 7.18821i 0.484343 − 0.726119i
\(99\) −0.368740 + 0.212892i −0.0370598 + 0.0213965i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 380.2.s.a.179.18 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.20 yes 112
5.4 even 2 inner 380.2.s.a.179.39 yes 112
19.12 odd 6 inner 380.2.s.a.259.37 yes 112
20.19 odd 2 inner 380.2.s.a.179.37 yes 112
76.31 even 6 inner 380.2.s.a.259.39 yes 112
95.69 odd 6 inner 380.2.s.a.259.20 yes 112
380.259 even 6 inner 380.2.s.a.259.18 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.18 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.20 yes 112 4.3 odd 2 inner
380.2.s.a.179.37 yes 112 20.19 odd 2 inner
380.2.s.a.179.39 yes 112 5.4 even 2 inner
380.2.s.a.259.18 yes 112 380.259 even 6 inner
380.2.s.a.259.20 yes 112 95.69 odd 6 inner
380.2.s.a.259.37 yes 112 19.12 odd 6 inner
380.2.s.a.259.39 yes 112 76.31 even 6 inner